
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm}   % This is your set screw

%%文档的题目、作者与日期
%\author{王立庆（2022级数学与应用数学1班）}
\author{ALEX }
\title{高等代数复习题 - 矩阵}
%\date{\vspace{-3ex}}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{2022 年 9 月 8 日}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %1
下列命题错误的是哪个？

\begin{enumerate}
\item  若干个初等矩阵的积必是可逆矩阵。
\item  可逆矩阵的和未必是可逆矩阵。
\item  两个初等矩阵的积仍是初等矩阵。
\item  可逆矩阵必是有限个初等矩阵的乘积。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %2
下列关于同阶可逆矩阵和不可逆矩阵的命题正确的是是哪个？

\begin{enumerate}
\item  两个不可逆矩阵之和仍是不可逆矩阵。
\item  两个可逆矩阵的和仍是可逆矩阵。
\item  两个不可逆矩阵之积必是不可逆矩阵。
\item  一个可逆矩阵和一个不可逆矩阵之积必是可逆矩阵。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %3
下列关于矩阵乘法交换性的结论错误的是哪个？

\begin{enumerate}
\item  若 $A$ 是可逆矩阵, 则 $A$ 与 $A^{-1}$ 可交换。
\item  可逆矩阵必与初等矩阵乘法可交换。
\item  任意 $n$ 阶矩阵与数量矩阵 $cE_n$ 乘法可交换, 这里 $c$ 是常数。
\item  初等矩阵与初等矩阵乘法未必可交换。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %4
设 $A$ 是可逆矩阵, 则下述哪个成立？

\begin{enumerate}
\item  $A$ 和任意同阶矩阵之积必是可逆矩阵。
\item  若 $B$ 是同阶初等矩阵, 则 $AB$ 的行列式不等于 0.
\item  若 $B$ 是同阶可逆矩阵, 则 $A+B$ 的行列式不等于 0.
\item  $A$ 和任意常数之积仍是可逆矩阵。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %5
设矩阵 $A$ 经过有限次初等变换后得到矩阵 $B$, 结论正确的是哪个？

\begin{enumerate}
\item  若 $A$ 和 $B$ 都是 $n$ 阶方阵, 则 $|A|=|B|$.
\item  若 $A$ 和 $B$ 都是 $n$ 阶方阵, 则 $|A|$ 与 $|B|$ 同时为 0 或同时不为 0.
\item  若 $A$ 是可逆矩阵, 则 $B$ 未必是可逆矩阵。
\item  $A=B$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %6
设 $A$ 是 $n$ 阶方阵, $A^*$ 是其伴随矩阵, 则下列结论错误的是哪个？

\begin{enumerate}
\item  若 $A$ 是可逆矩阵, 则 $A^*$ 也是可逆矩阵。
\item  若 $A$ 是不可逆矩阵, 则 $A^*$ 也是不可逆矩阵。
\item  若 $|A^*|\neq 0$, 则 $A$ 是可逆矩阵。
\item  $|AA^*|=|A|$.
\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %7
下列矩阵中可以化为有限个初等矩阵乘积的矩阵是哪个？

\begin{enumerate}
\item  $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 2 \end{pmatrix}$
\item  $\begin{pmatrix} 1 & 2 & 0 \\ 0 & -1 & 3 \\ 0 & 0 & 2 \end{pmatrix}$
\item  $\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{pmatrix}$
\item  $\begin{pmatrix} -1 & 2 & 3 \\ 0 & -2 & -1 \\ -3 & 2 & 7 \end{pmatrix}$
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %8 
关于初等矩阵，下述哪个说法是正确的？

\begin{enumerate}
\item  初等矩阵都可逆。
\item  初等矩阵相加仍是初等矩阵。
\item  初等矩阵的行列式值等于 1. 
\item  初等矩阵相乘仍是初等矩阵。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %9
设 $A=\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots& \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{pmatrix}$, 
$B=\begin{pmatrix}  A_{11} & \cdots & A_{1n} \\ \vdots& \ddots & \vdots \\ A_{n1} & \cdots & A_{nn} \end{pmatrix}$, 
其中 $A_{ij}$ 是 $a_{ij}$ 的代数余子式, 则正确的是哪个？

\begin{enumerate}
\item  $A$ 是 $B$ 的伴随矩阵。
\item  $B$ 是 $A$ 的伴随矩阵。
\item  $B$ 是 $A^T$ 的伴随矩阵。
\item  以上都不对。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %10
设 $A$ 是 $n$ 阶方阵，$B$ 是交换 $A$ 中两列所得到的方阵, 若 $|A|\neq |B|$, 则正确的是哪个？

\begin{enumerate}
\item  $|A|$ 可能为 0.
\item  $|A|\neq 0$.
\item  $|A+B| \neq 0$.
\item  $|A-B| \neq 0$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %11
设 $A, B, A+B$ 均为 $n$ 阶可逆矩阵, 则 $(A^{-1}+B^{-1})^{-1}$ 等于下述哪个？

\begin{enumerate}
\item  $A+B$.
\item  $A-B$.
\item  $(A+B)^{-1}$.
\item  $A(A+B)^{-1}B$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %12
如果矩阵 $\begin{pmatrix} 1 & a & 0 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{pmatrix}$ 是不可逆矩阵, 那么 $a$ 的值是多少？

\begin{enumerate}
\item  0.
\item  1.
\item  2.
\item  3.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %13
设 $A$ 和 $B$ 是 $3$ 阶矩阵, 且 $|A|=3, |B|=2$, 那么 $|2A^*B^{-1}|$ 等于多少？

\begin{enumerate}
\item  3.
\item  6.
\item  9.
\item  36.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%%\begin{eqnarray*}
%%\det(2A^*B^{-1})=\det(2|A|A^{-1}B^{-1}) =\det(6E)\det(A^{-1})\det(B^{-1})=(6)^3(3)^{-1}(2)^{-1}=36. 
%%\end{eqnarray*}
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %14
设 $A$ 是 4 阶方阵, $|A|=\frac{1}{2}$, 则 $|(2A)^{-1}+3A^*|$ 等于多少？

\begin{enumerate}
\item  4.
\item  16.
\item  32.
\item  64.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%%\begin{eqnarray*}
%%\det[(2A)^{-1}+3A^*] = \det\left(\frac{1}{2}A^{-1}+3|A|A^{-1} \right) = \det(2A^{-1}) = (2)^4\det(A)^{-1}=2^5. 
%%\end{eqnarray*}
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %15
设 $A=\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\  0 & a_{22}& \cdots & a_{2n} \\   \vdots & \vdots & \ddots & \vdots \\
   0 & 0 & \cdots & a_{nn} \end{pmatrix}$ 是上三角矩阵, 则 $A$ 可逆的充要条件是什么？

\begin{enumerate}
\item  $a_{11}\neq 0$. 
\item  $a_{22}\neq 0$. 
\item  $a_{nn}\neq 0$. 
\item  $a_{11}a_{22}\cdots a_{nn}\neq 0$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %16
设有分块上三角矩阵 $M=\begin{pmatrix}  A & B \\  O & C \end{pmatrix}$. 下述说法中，不正确的是哪个？

\begin{enumerate}
\item  当 $A$ 是不可逆矩阵时，$M$ 是不可逆矩阵。 
\item  当 $B$ 是不可逆矩阵时，$M$ 是不可逆矩阵。
\item  当 $C$ 是不可逆矩阵时，$M$ 是不可逆矩阵。
\item  当 $A,C$ 是可逆矩阵时，$M$ 是可逆矩阵。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %17
计算矩阵的乘法，结果矩阵的所有元素的和是多少？
\begin{eqnarray*}
\begin{bmatrix}1&2&3 \\ 4&5&6 \end{bmatrix}
\begin{bmatrix} 7&8&9 \\ 0&1&2 \\ 3&4&5 \end{bmatrix}. 
\end{eqnarray*}

\begin{enumerate}
\item  247.
\item  248.
\item  249.
\item  250.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %18
计算矩阵的乘法，结果矩阵的所有元素的和是多少？
\begin{eqnarray*}
\begin{bmatrix} 1&2 \\ 3&4  \end{bmatrix}
\begin{bmatrix} 5&6 \\ 7&8  \end{bmatrix}
\begin{bmatrix} 9&0 \\ 1&2  \end{bmatrix}.
\end{eqnarray*}

\begin{enumerate}
\item  774.
\item  775.
\item  776.
\item  777.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %19
计算矩阵的乘法，结果矩阵的所有元素的和是多少？
\begin{eqnarray*}
\begin{bmatrix}1&n \\ 0&1 \end{bmatrix}
\begin{bmatrix} 1&m \\ 0&1 \end{bmatrix}. 
\end{eqnarray*}

\begin{enumerate}
\item  $1+n+m$. 
\item  $2+n+m$. 
\item  $3+n+m$. 
\item  $4+n+m$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %20
计算矩阵的乘法，结果矩阵的所有元素的和是多少？
\begin{eqnarray*}
\begin{bmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha &\cos\alpha \end{bmatrix}
\begin{bmatrix}\cos\beta & -\sin\beta \\ \sin\beta &\cos\beta \end{bmatrix}.
\end{eqnarray*}

\begin{enumerate}
\item  $2\cos(\alpha+\beta)$. 
\item  $2\cos(\alpha-\beta)$. 
\item  $2\sin(\alpha+\beta)$. 
\item  $2\sin(\alpha-\beta)$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %21
设矩阵 $A=\begin{pmatrix}1&2&3 \\ 4&5&6 \end{pmatrix}$, 
矩阵 $B= \begin{pmatrix} 7&8&9 \\ 0&1&2 \\ 3&4&5 \end{pmatrix}$. 
将矩阵 $A$ 与 $B$ 分别按行与按列分块，写成 $A=\begin{pmatrix}\alpha_1 \\ \alpha_2 \end{pmatrix}$
与 $B=(\beta_1, \beta_2, \beta_3)$ 的形式。
%
下述说法中，不正确的是哪个？

%分别计算 $A\beta_1$, $A\beta_2$, $A\beta_3$, 与 $\alpha_1 B$, $\alpha_2 B$.  将结果与直接计算 $AB$ 的结果进行比较。

\begin{enumerate}
\item  $AB=(A\beta_1, A\beta_2, A\beta_3)$. 
\item  $AB=\begin{pmatrix} \alpha_1 B \\ \alpha_2 B \end{pmatrix}$. 
\item  $A^t = (\alpha_1^t, \alpha_2^t)$. 
\item  $B^t = (\beta_1^t , \beta_2^t, \beta_3^t)$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %22
记 $E_{ij}$ 是第 $(i,j)$ 元素为 1 而其余元素为 0 的四阶矩阵。
%这里 $1\le i,j\le 4$. 
%\begin{enumerate}
%\item  分别计算 $E_{23}E_{34}$ 与 $E_{34}E_{12}$.
%\item  分情况讨论，计算 $E_{ij}E_{k\ell}$.
%\end{enumerate}
下述说法中，不正确的是哪个？

\begin{enumerate}
\item  $E_{12}E_{21}=E_{11}$. 
\item  $E_{23}E_{32}=E_{22}$. 
\item  $E_{12}E_{14}=E_{24}$. 
\item  $E_{23}E_{33}=E_{23}$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %23
如果矩阵 $A$ 与所有的 $n$ 阶矩阵的乘法都可以交换，那么 $A$ 是什么样的矩阵？

\begin{enumerate}
\item   $n$ 阶对角矩阵。
\item   $n$ 阶上三角矩阵。
\item   $n$ 阶下三角矩阵。
\item   $n$ 阶数量矩阵。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %24
设 $A$ 与 $B$ 是任意的两个 $n$ 阶矩阵。记 $E$ 是 $n$ 阶单位矩阵。
下述等式中，总是成立的是哪个？

\begin{enumerate}
\item  $(A-B)(A+B)=A^2-B^2$. 
\item  $(A-B)(A^2+AB+B^2)=A^3-B^3$. 
\item  $(A+B)(A^2-AB+B^2)=A^3+B^3$. 
\item  $(E-A)(E+A+A^2)=E-A^3$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %25
当 $B=B^t$ 时称 $B$ 为对称阵，当 $C=-C^t$ 时称 $C$ 为反对称阵。其中 $(\cdot)^t$ 是矩阵的转置运算。
将矩阵 $A=\begin{bmatrix}1&2&3 \\ 4&5&6 \\ 7&8&9 \end{bmatrix}$ 写成一个对称阵 $B$ 与一个反对称阵 $C$ 的和。
则矩阵 $B$ 的第 $(2,3)$ 元素是多少？
%\begin{enumerate}
%\item  写出三阶对称阵与反对称阵的一般形式。
%\item  将矩阵 $A=\begin{bmatrix}1&2&3 \\ 4&5&6 \\ 7&8&9 \end{bmatrix}$ 写成一个对称阵与一个反对称阵的和。
%\item  证明任意 $n$ 阶矩阵都能写成一个对称阵与一个反对称阵的和。
%\end{enumerate}

\begin{enumerate}
\item  3. 
\item  5. 
\item  7. 
\item  9. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %26
设矩阵 $A=\begin{bmatrix}1&2 \\ 3&0 \end{bmatrix}$, 设多项式 $f(x)=x^3+2x^2+3x+4$. 
则矩阵 $f(A)$ 的所有元素的和是多少？

\begin{enumerate}
\item  116. 
\item  117. 
\item  118. 
\item  119. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %27
设矩阵 $A$ 和 $C$ 是 $n$ 阶矩阵，设 $C$ 可逆。设 $f(x)$ 是一个多项式. 证明 $f(CAC^{-1})=Cf(A)C^{-1}$.
下述步骤中，有问题的是哪一步？
\begin{enumerate}
\item  设 $f(x)=a_nx^n+\cdots+a_1x+a_0$. 
\item  计算 $f(CAC^{-1})=a_n(CAC^{-1})^n+\cdots+a_1(CAC^{-1})+a_0E$.
\item  计算 $(CAC^{-1})^n=(CAC^{-1})(CAC^{-1})\cdots (CAC^{-1}) = C^nA^nC^{-n}$. 
\item  计算 $Cf(A)C^{-1} = C(a_nA^n+\cdots+a_1A+a_0E)C^{-1}$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %28
设矩阵 $A,B,C$ 是 $n$ 阶矩阵。记 $[A,B] = AB - BA$. 则下述哪个等式不一定成立？
\begin{enumerate}
\item  $[B,A] = -[A,B]$.
\item  $[A,BC] = [A,B]C+B[A,C]$.
\item  $[A,B+C] = [A,B] + [A,C]$.
\item  $[[A,[B,C]] = [[A,B],C]$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %29
对四阶单位矩阵依次进行下述初等变换。则得到的矩阵的所有元素的和是多少？
\begin{enumerate}
\item[(1)]  把第一行与第四行交换。
\item[(2)]  把第二行的3倍加到第三行。
\item[(3)]  把第二列的2倍加到第三列。
\end{enumerate}

\begin{enumerate}
\item  7.
\item  8.
\item  9.
\item  10.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %30
记 $T_{ji}(k)$ 是将第 $i$ 行乘以 $k$ 加到第 $j$ 行对应的初等矩阵，
$D_i(c)$ 是将第 $i$ 行乘以 $c$ 对应的初等矩阵，
$P_{ij}$ 是交换第 $i$ 行与第 $j$ 行对应的初等矩阵。 
设矩阵 $A=\begin{bmatrix}1&2 \\ 3&4 \end{bmatrix}$.
通过一些行初等变换，将 $A$ 化为单位矩阵。
下述哪个做法是正确的？
%\begin{enumerate}
%\item  通过一些行初等变换，将 $A$ 化为单位矩阵。
%\item  通过一些列初等变换，将 $A$ 化为单位矩阵。
%\end{enumerate}

\begin{enumerate}
\item  $T_{12}(1)D_2(-1/2)T_{21}(-3)A=E$. 
\item  $D_2(-1/2)T_{12}(1)T_{21}(-3)A=E$. 
\item  $T_{21}(-3)D_2(-1/2)T_{12}(1)A=E$. 
\item  $D_2(-1/2)T_{21}(-3)T_{12}(1)A=E$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %31
设矩阵 
%\begin{eqnarray*}
$A=\begin{bmatrix}2&1&3 \\ 3&1&2 \\ 2&3&1 \end{bmatrix}$,
%\end{eqnarray*}
计算伴随矩阵 $A^*$ 的所有元素的和，与逆阵 $A^{-1}$ 的所有元素的和。

\begin{enumerate}
\item  $6, 1/2$. 
\item  $6, 1/3$.
\item  $9, 1/2$. 
\item  $9, 1/3$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %32
记 $\omega=\exp(2\pi i/3)$. 设矩阵 
%\begin{eqnarray*}
$A=\begin{bmatrix}1&1&1 \\ 1& \omega & \omega^2 \\ 1& \omega^2 & \omega \end{bmatrix}$. 
%\end{eqnarray*}
则这个矩阵的伴随矩阵的所有元素的和是多少？

\begin{enumerate}
\item  $3\omega^2 - 3\omega$. 
\item  $3\omega^2 + 3\omega$. 
\item  $3\omega - 3$. 
\item  $3\omega + 3$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%%\begin{eqnarray*}
%%A^*=\begin{bmatrix} \omega^2-\omega & \omega^2-\omega &\omega^2-\omega \\ 
%%\omega^2-\omega & \omega-1 & 1-\omega^2  \\ 
%%\omega^2-\omega & 1-\omega^2 & \omega-1 
%%\end{bmatrix}. 
%%\end{eqnarray*}
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %33
计算下述矩阵的的逆矩阵的第 $(1,2)$ 元素，
$%\begin{eqnarray*}
\begin{bmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha &\cos\alpha \end{bmatrix}.
$%\end{eqnarray*}

\begin{enumerate}
\item  $\sin\alpha$. 
\item  $-\sin\alpha$.
\item  $\cos\alpha$.
\item  $-\cos\alpha$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %34
设矩阵 $A=\begin{bmatrix}1&-1&2&-3 \\ 0&1&-1&2 \\ 0&0&1&-1 \\ 0&0&0&1 \end{bmatrix}$. 记 $E$ 是 4 阶单位矩阵。使用下述四种方法求逆阵。
\begin{enumerate}
\item[(1)]  用初等变换的方法求 $A$ 的逆阵。
\item[(2)]  用伴随矩阵的方法求 $A$ 的逆阵。
\item[(3)]  将矩阵 $A$ 写成 $E-B$ 的形式，验证 $(E-B)(E+B+B^2+B^3)=E$.
\item[(4)]  用分块矩阵的方法求 $A$ 的逆阵。
\end{enumerate}
矩阵 $A$ 的逆阵的第 $(1,4)$ 元素是多少？

\begin{enumerate}
\item  $1$. 
\item  $-1$. 
\item  $0$. 
\item  $3$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%%逆阵是 $A^{-1}=\begin{bmatrix}1&1&-1&0 \\ 0&1&1&-1 \\ 0&0&1&1 \\ 0&0&0&1 \end{bmatrix}$.
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %35
将矩阵 $A=\begin{bmatrix} 0&1&1 \\ 1&0&1 \\ 1&1&0 \end{bmatrix}$ 写成一些初等矩阵的乘积。
下述哪个是正确的？

\begin{enumerate}
\item  $A= T_{13}(-1) T_{23}(-1) D_3(-1/2) T_{32}(-1) T_{31}(-1) P_{12}$. 
\item  $A = P_{12} T_{31}(1) T_{32}(1) D_3(-2) T_{23}(1) T_{13}(1)$. 
\item  $A= T_{13}(-1) T_{23}(-1) D_3(1/2) T_{32}(-1) T_{31}(-1) P_{12}$. 
\item  $A = P_{12} T_{31}(1) T_{32}(1) D_3(2) T_{23}(1) T_{13}(1)$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %36
将矩阵 $A=\begin{bmatrix} 2&3 \\ 3&5 \end{bmatrix}$ 写成一些第三类初等矩阵的乘积。
下述哪个是正确的？

\begin{enumerate}
\item  $A = T_{12}(1/3) T_{21}(3) T_{12}(4/3)$. 
\item  $A = T_{12}(-1/3) T_{21}(-3) T_{12}(-4/3)$. 
\item  $A = T_{12}(4/3) T_{21}(3) T_{12}(1/3)$. 
\item  $A = T_{12}(-4/3) T_{21}(-3) T_{12}(-1/3)$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %37
用初等变换求下述矩阵的逆阵，
$%\begin{eqnarray*}
A = \begin{bmatrix}1&1&1 \\ 0&1&1 \\ 0&0&1 \end{bmatrix}. 
$%\end{eqnarray*}
矩阵 $A$ 的逆阵的第 $(1,3)$ 元素是多少？

\begin{enumerate}
\item  $0$. 
\item  $1$. 
\item  $-1$. 
\item  $2$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%%\begin{eqnarray*}
%%A^{-1} = \begin{bmatrix}1&-1&0 \\ 0&1&-1 \\ 0&0&1 \end{bmatrix}. 
%%\end{eqnarray*}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %38
用初等变换求下述矩阵的逆阵，
$%\begin{eqnarray*}
A = \begin{bmatrix} 2&5&7 \\ 6&3&4 \\ 5&-2&-3 \end{bmatrix}. 
$%\end{eqnarray*}
矩阵 $A$ 的逆阵的第 $(3,3)$ 元素是多少？

\begin{enumerate}
\item  $22$. 
\item  $23$.
\item  $24$. 
\item  $25$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%%\begin{eqnarray*}
%%A^{-1} = \begin{bmatrix}1&-1&1 \\ -38&41&-34 \\ 27&-29&24 \end{bmatrix}. 
%%\end{eqnarray*}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %39
用初等变换求下述矩阵的逆阵，
$%\begin{eqnarray*}
A = \begin{bmatrix} 1&1&0 \\ 0&1&0 \\ 0&3&3 \end{bmatrix}.
$%\end{eqnarray*}
矩阵 $A$ 的逆阵的第 $(3,3)$ 元素是多少？

\begin{enumerate}
\item  $1$.
\item  $1/2$.
\item  $1/3$.
\item  $1/4$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%%\begin{eqnarray*}
%%A^{-1} = \begin{bmatrix}1&-1&0 \\ 0&1&0 \\ 0&-1&1/3 \end{bmatrix}. 
%%\end{eqnarray*}
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %40
求解下述矩阵方程，
$%\begin{eqnarray*}
\begin{bmatrix}1&3 \\ 1&2 \end{bmatrix}X=\begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix}. 
$%\end{eqnarray*}
矩阵 $X$ 的所有元素的和是多少？

\begin{enumerate}
\item  $1$. 
\item  $2$. 
\item  $3$. 
\item  $4$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%%\begin{eqnarray*}
%%X=\begin{bmatrix} 1&1 \\ 0&0 \end{bmatrix}. 
%%\end{eqnarray*}
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %41
求解下述矩阵方程，
$%\begin{eqnarray*}
X\begin{bmatrix}-1&1 \\ 3&-4 \end{bmatrix}=\begin{bmatrix} -2&-1 \\ 3&4 \end{bmatrix}. 
$%\end{eqnarray*}
矩阵 $X$ 的所有元素的和是多少？

\begin{enumerate}
\item  $-17$. 
\item  $-18$. 
\item  $-19$. 
\item  $-20$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%%\begin{eqnarray*}
%%X=\begin{bmatrix} 11&3 \\ -24&-7 \end{bmatrix}. 
%%\end{eqnarray*}
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %42
求解下述矩阵方程，
$%\begin{eqnarray*}
\begin{bmatrix}3&1 \\ 2&1 \end{bmatrix}X\begin{bmatrix}1&3 \\ 1&2 \end{bmatrix}=\begin{bmatrix} 3&3 \\ 2&2 \end{bmatrix}.
$%\end{eqnarray*}
矩阵 $X$ 的所有元素的和是多少？

\begin{enumerate}
\item  $1$. 
\item  $2$. 
\item  $3$. 
\item  $4$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(a). 
%
%%\begin{eqnarray*}
%%X=\begin{bmatrix} -1&2 \\ 0&0 \end{bmatrix}. 
%%\end{eqnarray*}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %43
记 $A^*$ 是 $n$ 阶矩阵 $A$ 的伴随矩阵，证明 $\det(A^*) = \det(A)^{n-1}$. 这里 $\det(\cdot)$ 是求行列式的值。
下述哪一步是不正确的？

\begin{enumerate}
\item  由伴随矩阵的性质可得 $AA^* = dE$, 这里记 $d=\det(A)$. 
\item  两边求行列式的值，可得 $\det(AA^*)=\det(dE)=d^n$. 
\item  由行列式乘积公式，可得 $\det(AA^*) = \det(A)\det(A^*)$. 
\item  由 $\det(A)\det(A^*) = d^n$ 可得 $\det(A^*) = d^{n-1}$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(d). 
%
%%矩阵 $A$ 的行列式的值可能等于零。这时候就不能从等式两边约去 $d$ 了。
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %44
设 $A$ 和 $B$ 都是 $n$ 阶矩阵，证明若 $AB$ 可逆，则 $A$ 和 $B$ 都可逆。
下述哪一步是不正确的？

\begin{enumerate}
\item  因为 $AB$ 可逆，所以存在矩阵 $C$ 使得 $(AB)C=E$. 
\item  根据矩阵乘法的交换律，从 $(AB)C=E$ 可得 $A(BC)=E$. 
\item  从 $A(BC)=E$ 可得 $BC$ 是矩阵 $A$ 的逆阵。
\item  从 $(BC)A=E$ 可得 $B(CA)=E$, 所以 $CA$ 是 $B$ 的逆阵。 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(b). 
%
%%根据矩阵乘法的结合律，从 $(AB)C=E$ 可得 $A(BC)=E$. 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %45
设 $A$ 和 $B$ 都是 $n$ 阶矩阵，证明若 $AB=E$, 则 $BA=E$, 从而 $A$ 和 $B$ 互为逆阵。
下述哪一步使用了线性方程组有解的判别定理？

\begin{enumerate}
\item  从 $AB=E$ 可得 $\det(AB)=\det(E)=1$. 
\item  从 $\det(AB)=1$ 可得 $\det(A)\det(B)=1$. 因此有 $\det(A)\neq 0$ 与 $\det(B)\neq 0$.  
\item  从 $\det(B)\neq 0$ 可得存在 $n$ 阶矩阵 $X$ 使得 $BX=E$.  
\item  从 $A=AE=A(BX)=(AB)X=EX=X$ 可得 $X=A$. 因此 $BA=E$. 
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red} 解答：(c). 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
